"by the 3rd or 4th century AD" - that early?!
Aramaic lived for a longer period, but then slowly died out. As a liturgical language western Aramaic / Syriac lived on, and eastern variants for longer but as a widespread, spoken lived language in the Levant I believe Greek replaced it, and then Arabic. I understand that "Jewish Palestinian Aramaic" lingered on for some time in some smaller communities.
I think that was only the case in urban areas and/or amongst the more educated upper and middle classes. Syriac and other Aramaic languages were still widespread amongst local Christian population well into the Islamic perod.